Your problem will be visa issues.
If you were accepted on a graduate programme you would get a student visa. This would enable you to work part-time (up to 20 hours a week). Realistically this wouldn't enable you to earn enough money to actually fund you studies, and probably you would have to pay tuition fees in advance anyway. So if you go this route, you need to be able to pay, or find a scholarship (usually originating in the US. UK schools generally do not offer financial aid to overseas students. There is a bit more for graduate students).
Just turning up and looking for a job is a bad idea! You would not be elligible to work if you entered the country as a tourist. The easiest way to get a work visa would be to arrange employment before you leave the US. Then your employer will help you get a visa (do NOT just turn up and tell immigration you are looking for work......).
There are also some schemes which help you get an internship in the UK short term, such as BUNAC which I have linked below.
Intern in Britain
This is the teach first programme which was discussed above.
Teach First
I am not sure if you are elligible for it because you have not graduated from a UK univeristy. Also, I am not sure that a UK teaching qualification will be recognised in any way in the US. I would email them and ask before applying.
If you just want to volunteer, and won't be paid, you can enter the UK for up to 6 months with no visa.
If you are in some way elligible for UK citizenship (or in fact citizenship of most - but not all - EU countries. I think there are some issues if you are Romanian/Bulgarian) apply for a passport and your problems will be over. It sounds like you might be if your grandparenst were British. You will be able to stay/study/work without restriction. This also opens up a whole world of generously funded PhD programmes as well (it did for me!). However, there are a LOT more places in sciences as opposed to arts.